Welcome to the LAB component of Math 3046. Lets
assume that you are logged into your personal computer or a Temple University
(TU) computer. Now start MATLAB and Microsoft
Word (or equivalent word processor. If you are on your own computer you know
where the programs are located. If you are on a TU computer you can locate these
programs from an icon that may appear on the desk top or if you are running
Windows on the desktop click here for
directions. (If you are on a MAC you are on your own!)

In Word create a new document.
Put your full name and TU ID at the top of the document.
As you work through a lab assignment you should enter all the commands that are
listed into MATLAB and work through the Examples.
But
in
your
Word document you only need to include your responses to
the Exercises
together with the name of the lab, the number
of the Exercise, and your answer. Your answer can a
combination of the following four items you are answering. Clearly label
any parts to an exercise and use a row of plus signs (++++++++++++++++++)
between answers to exercises.

Credits: This design and approach was adapted from work at
University of California at San Diego by Bill Helton and a host of contributors.

Icon:

Item:

Comments:

Commands entered in MATLAB & resulting
output

You should copy relevant input and output from
MATLAB and paste it into your Word document. You need only include commands that worked.

Plots & graphs

Include all graphs generated in an exercise
unless the problem specifically tells you which/how many to include.

Full sentence response

Exercise contains a question that you need to
use at least one or two complete sentences to answer. Even if
you're stuck, write down any reasoning or ideas you've had.

Requires work by hand

Do scratch work by hand. Leave space in
your Word document and write your scratch work directly on the
assignment to turn in.

►Graphing Functions◄

MATLAB provides several methods for plotting the graphs of functions and other
curves. The easiest to use is what we will call EZ plotting, since it uses the
command ezplot and its variants. While EZ plotting is easy to use it is not as
flexible as several other methods which are available.

Example 1. (EZ plotting)

To plot a function with the ezplotcommand we first need to form an expression for the function.

>> ezplot('cos(x)/(1+x^2)')
%Note that the expression is included between single quotes. Result is Figure 1.

Figure
1.

Figure 2.

Figure 3.

In Figure 1 the domain was selected automatically and a title
appears at the top. MATLAB used the default domain of \(\left[ { - 2\pi ,\,2\pi
} \right]\). We can change the domain used by specifying it as part
of the ezplotcommand as shown next. This is shown
in Figure 2. It is often the case that our first attempt when we choose the
domain is not completely satisfactory. Do not be afraid to redo a plot to make
it look better.

>> ezplot('cos(x)/(1+x^2)',
[-9,9]) %Result is Figure 2.

>> ezplot('cos(x)/(1+x^2)',
[-9,9,-1,1]) %Result is Figure 3.

In Figure 3 the domain was specified as [-9, 9] and the range is [-1,
1]. You may need to make several adjustments to the domain and range to obtain the picture you want. You
need not specify a range but then you may not have all of the curve displayed as
illustrated in Figure 2. Since another example follows we will delete any
current figures in MATLAB using the following command.

>> close all %Delete all active
figures.

Example 2. (Plotting usingfplot.)

To plot a function with the fplot
command we first need to form an expression for the function as with
ezplot. This time we
give the function to plot a name using the inline
command. Again the expression must be between single quotes.

>> g = inline('cos(x)/(1+x^2)')

Now we want to graph g(x) on the
interval [-10, 10]. See Figure 4.

>> fplot(g, [-10, 10])
>> title('An fplot
graph.')

Figure
4.

MATLAB has the result of the previous commands in a
figure window. If another ezplot or
fplot command is executed then the current figure
will be replaced by the new figure. The command figure
brings up a new graphics window. MATLAB will draw on whatever graphics
window was most recently in the front.

>> figure
>> ezplot(g, [-5, 5])

Now there are two active graphics windows, Figures 1 and 2.
Commands figure(1) and
figure(2) will display the two graphs.

Exercise 1.(a)Graph the function \(f(x) = e^{0.1x} \cos (2x)\)
on the interval [0, 5] with ezplot.
Include the MATLAB code you used. Then paste the graph into your
Word document. To copy the graph bring its window to the front (if
it is not visible use command shg)
and choose the menu option Edit => Copy
Figure. Include a
sentence describing the \[\mathop {\lim }\limits_{x \to \infty } f(x).\]
(b) The initial value problem \(y' = 2y - t,\,\,y(0) = - 1\) has
solution \(y(t) = \frac{t}{2} - \frac{5}{4}e^{2t} + \frac{1}{4}\). Graph
y(t) on the interval [0, 1] using fplotand put a title on the graph that is your
name. Include the MATLAB code you used. Then paste the graph into
your Word document. To copy the graph bring its window to the
front (if it is not visible use command shg)
and choose the menu option Edit => Copy
Figure. Include a sentence describing the
\[\mathop {\lim }\limits_{t \to \infty } y(t).\]
(c) Graph the function \(g(x) = e^{ - 0.1x}
\) on the interval [0, 8] with
ezplot. Next use
commandhold on,
which retains the graphics window so we can plot a second function in
the same window. Graph \(f(x)
= e^{-0.1x} \sin (2x)\) on the interval [0,
8] and put a title on the graph that is your
name. Include the MATLAB code you used. Then paste
the graph into your Word document. To copy the graph bring its
window to the front (if it is not visible use command shg) and choose the
menu option Edit => Copy Figure.
Once you have things in the word file use the command hold off
which releases the graphics screen so that a new plot can replace the
current plot. Finally Include a
sentence describing the \[\mathop {\lim }\limits_{x \to \infty } f(x).\]

Plotting using the plot command.

Previously when we used the ezplot
or fplot command we constructed an expression for a
function and used the expression as input to the commands. The plot command
requires we first construct a set of ordered pairs before using the command.
Then points corresponding to the ordered pairs can be displayed using various
symbols and connected with a variety of solid or dashed lines. So instead of
using a function expression the plot command
employs a discrete set of points hence there is more flexibility, but at the
cost some ease of use. The set of ordered pairs can be constructed as two arrays
or vectors of data, say x-values and y-values.

Example 3.(Plotting a set of ordered pairs.)

A set of ordered pairs can be constructed by choosing x-values
and the same number of y-values. There need not be a function involved. In
MATLAB an array or vector is constructed by including a list of numerical values
between square brackets with the entries separated by a space or comma. Each of
the following commands includes a variation of the plot command.

Graph the function \(f(x) = 3x^2 - 2x^3 \) over interval [-1,
3] by using a set of equispaced points from x = -1 to x = 3. To generate the
sets of ordered pairs we will illustrate two techniques. To define a vector that
contains a large number of equispaced points in [-1, 3] we use MATLAB's colon
operator in the form start: increment : finish, where start is
-1, finish is 3, and we choose the increment.

For small increments we can use a semicolon after the finish
value. To get a smooth graph of f(x) we will use the increment 0.05 which gives
81 values in [-1, 3]. To get the corresponding values of f(x) we employ MATLAB's
vector oriented capability. That is, evaluate the function at the vector x
instead of each entry. To this end the standard arithmetic operators are
replaced by corresponding vector operators. To evaluate f(x) at a single value
x the MATLAB command y = 3*x^2-2*x^3 is sufficient.
If we precede each arithmetic operator by a period we invoke the vector
evaluation. This command is y = 3.*x.^2-2.*x.^3
which generate a set of 81 y-values to use with plot.

Inserting vector operators tends to be error prone, so
MATLAB has a command that performs the task. The command is named vectorize. For details use
help vectorize. Once
the function is vectorized we still need to evaluate it so we use the eval command.

Graph \(f(x) = \frac{{4\sin (3x) - 5\cos (x)}}{{e^x }}\)
on [0, 5] using 200 equispaced points. To generate the points we use command linspace which has the format linspace(X1, X2, N) to generate N points between values X1
and X2.

>> f='(4*sin(2*x)-5*cos(x))/exp(x)';x=linspace(0,5,200);y=eval(vectorize(f));
plot(x,y,'d:r') %Copy this command and execute in MATLAB.

The commands used above include

ezplot
fplot
plot

:operator instart:
increment : finish
linspace
vectorizeeval

shg
hold on
hold off
title

Exercise 2.(a)Graph the function \(f(x) = \frac{{e^x }}{{x^2 +
1}}\)
on the interval [-3, 3] with plot.
Use linspace with N = 50, the vectorize,
and eval commands, connect the points, indicate the points with an
asterisk and use c for the color; use 'c*-'. Include the MATLAB code you used. Then paste the graph into your
Word document. To copy the graph bring its window to the front (if
it is not visible use command shg)
and choose the menu option Edit => Copy
Figure.

(b) The initial value problem \(y' = 2y - t,\,\,y(0) = - 1\) has
solution \(y(t) = \frac{t}{2} - \frac{5}{4}e^{2t} + \frac{1}{4}\). Graph
y(t) on the interval [0, 1] using plot,
construct t-values with
increment 0.05,
use the color red, and put a title on the graph that is your
name. Include the MATLAB code you used. Then paste the graph into
your Word document. To copy the graph bring its window to the
front (if it is not visible use command shg)
and choose the menu option Edit => Copy
Figure.

(c) The height
h(t) and horizontal distance x(t) traveled by a ball thrown at angle α
with an initial velocity v are given by
\[h(t) = v\,t\sin \left( \alpha \right) - \frac{1}{2}gt^2 ,\,\,x(t) =
v\,t\,\cos \left( \alpha \right).\]We take \(g = 9.81m/s^2 \) and suppose the ball is thrown with velocity
\(v = 15m/s\) at an angle of \(\pi /6\) radians. Determine how long the
projectile is in flight; call that value tmax. Then use
plot to graph a smooth
curve for h(t) and x(t) over [0, tmax]. Include the MATLAB code you used. Then paste the graphs into your
Word document. To copy the graph bring its window to the front (if
it is not visible use command shg)
and choose the menu option Edit => Copy
Figure.

(d) Graph the function \(f(x) = \left| {\cos (x)} \right|\) on interval
[0, 2π]
using 500 equispaced points. After you obtain the graph use the command
grid in
MATLAB. Include the MATLAB code you used. Then paste the graph into your
Word document. To copy the graph bring its window to the front (if
it is not visible use command shg)
and choose the menu option Edit => Copy
Figure. Explain what the grid command did.